Theorem riotasv2s 38940

Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5409) in the form of a substitution instance. Special case of riota2f 7412. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypothesis

Ref	Expression
riotasv2s.2	⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))

Assertion

Ref	Expression
riotasv2s	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s

Step	Hyp	Ref	Expression
1		3simpc 1149	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → (𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)))
2		simp1 1135	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐴 ∈ 𝑉)
3		riotasv2s.2	. . . . . 6 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
4		nfra1 3282	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
5		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
6	4, 5	nfriota 7400	. . . . . 6 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
7	3, 6	nfcxfr 2901	. . . . 5 ⊢ Ⅎ𝑦𝐷
8	7	nfel1 2920	. . . 4 ⊢ Ⅎ𝑦 𝐷 ∈ 𝐴
9		nfv 1912	. . . . 5 ⊢ Ⅎ𝑦 𝐸 ∈ 𝐵
10		nfsbc1v 3811	. . . . 5 ⊢ Ⅎ𝑦[𝐸 / 𝑦]𝜑
11	9, 10	nfan 1897	. . . 4 ⊢ Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)
12	8, 11	nfan 1897	. . 3 ⊢ Ⅎ𝑦(𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑))
13		nfcsb1v 3933	. . . 4 ⊢ Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶
14	13	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶)
15	10	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
16	3	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
17		sbceq1a 3802	. . . 4 ⊢ (𝑦 = 𝐸 → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
18	17	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
19		csbeq1a 3922	. . . 4 ⊢ (𝑦 = 𝐸 → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
20	19	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
21		simpl 482	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 ∈ 𝐴)
22		simprl 771	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐸 ∈ 𝐵)
23		simprr 773	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
24	12, 14, 15, 16, 18, 20, 21, 22, 23	riotasv2d 38939	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝐴 ∈ 𝑉) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
25	1, 2, 24	syl2anc 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888  ∀wral 3059  [wsbc 3791  ⦋csb 3908  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-undef 8297

This theorem is referenced by: (None)